Imperfect Labor and Products Markets,

Regulation,

and the Union Wage Gap

Helge Sannersanner@rz.uni-potsdam.de

Department of Economics and Social Sciences

University of Potsdam

November 16, 2005

1 Introduction

This paper presents a theoretical model designed to capture the most important

interactions between imperfect product and labor markets. This issue has attracted

a great deal of attention recently (see e.g. Nickell (1999), Nicoletti, Bassanini,

Ernst, Jean, Santiago and Swaim (2001), and Ebell and Haefke (2003)). The reason

is, as Nicoletti et al. (2001, p. 7) put it: “... accounting for the potential cross-

market effects of product and labor market policies appears to be an important

element of good policy design.” The main difference of our approach to other

recently published contributions (see in particular Blanchard and Giavazzi, 2003)

is that we consider a dual labor market. We think that the inclusion of a secondary

labor market not only increases realism, but also heals some of the logical problems

of related literature. In addition, the dual labor market renders possible to calculate

the ’union wage gap’, i.e. the difference between the pay of homogeneous workers

that are / are not covered by collective bargaining. Through this, we are able

provide an explanation of the union wage gap, which previous literature could not

explain satisfactorily.

The union wage gap derived from the model for various countries also serves as

a check for the appropriateness of our model. By comparing our results to estimates

from the empirical literature, we are able to verify that our approach indeed seems

to account for the most important forces at work. If one accepts that the model

reflects the basic interactions between imperfect markets, it is straightforward to

ask how economic policy may influence the outcome in the model. We chose two

parameters, one for the goods markets and one for the labor market, that may

(partially) be determined by policy measures, and investigate how these parameters

may be used to foster real income.

The following section introduces a simple general equilibrium model of imper-

fect product and labor markets. In section 3 we derive the union wage gap for some

industrialized countries numerically. Section 4 identifies parameters that may be

influenced by regulation / deregulation policy, and analyzes how variations of these

parameters impact on the size and distribution of rents. Section 5 concludes.

1

2 A model of imperfect in- and output markets

Imagine a two-sector economy. One monopolistic competitive sector, producing

heterogeneous goods with increasing returns to scale, and one perfectly compet-

itive sector producing a homogeneous good with constant returns to scale. This

setup has recently been supported by empirical work on scale elasticities. After

an examination of trade data from 71 countries, Antweiler and Trefler (2002) con-

clude: “Our results point to the importance of integrating constant- and increasing-

returns-to-scale industries within a single general-equilibrium framework.”

The market for labor is dichotomized as well. Some workers receive the com-

petitive wage rate, and some workers receive the (higher) union wage rate. While

firms in the homogeneous sector employ exclusively workers from the competitive

labor market, production in the heterogeneous sector requires unionized labor as an

input. Although in reality labor markets consist of more than only two sectors, the

simplifying assumption of a dual labor market is supported by empirical evidence

(see the survey in Saint-Paul (1996)).

Firms in the heterogeneous sector each employ a fixed amount of non-unionized

labor. Wages paid to these workers have the character of fixed costs, because the

competitive wage rate is fully determined by the technology of the homogeneous

sector. In our interpretation these costs arise e.g. due to security agents, cleaner,

gate keeper, and all other services, who are essential for the firm to produce goods,

but whose number is yet independent of the amount actually produced. Equiv-

alently, the fixed labor input may be seen as the corresponding amount of the

homogeneous good.

2.1 Workers

There are N homogeneous workers, indexed by j. Utility of a worker depends on

the consumption of one homogeneous and n heterogeneous goods (x0,j, and xi,j

with i ∈ {1, 2, . . . , n}, respectively). The utility function of a representative worker

is

uj = u(x0,j, x1,j, . . . , xn,j) = x1−β0,j ·Xβ

j (1)

2

with

β ∈ (0, 1),

Xj ≡

(n−(1−ρ)

n∑i=1

xi,jρ

) 1ρ

,

ρ ∈ (0, 1)

where x0,j stands for consumption of the homogeneous good, β symbolizes the

expenditure share, and Xj is a composite index of the consumed varieties (see

Dixit and Stiglitz (1977) and Blanchard and Giavazzi (2003)). ρ corresponds with

the elasticity of substitution, σ, according to the definition σ ≡ 1/(1− ρ) > 1. In

comparison to the original Dixit and Stiglitz-approach, ρ is derived endogenously

through the assumed relationship

ρ = 1− 1/(ζn), ζn = σ > 1 (2)

where the exogenous parameter ζ determines how strong ρ and the elasticity of sub-

stitution between any two varieties depend on the number of firms.1 One possible

interpretation why a higher number of firms increases the elasticity of substitution

is that the varieties become closer substitutes. We see ζ as a proxy for the degree

of transparency on the products market. It is necessary that changes of the supply

structure be transparent to the consumers for market entry to have an impact on

consumer behavior.

Apart from ρ being endogenous the main difference to the Dixit-Stiglitz frame-

work is the term n−(1−ρ) in the definition of the composite index X. The effect

of this term becomes clear when we assume for the moment that consumption of

each heterogeneous variety is the same, i.e. x1,j = x2,j = . . . = xj. In this case we

get Xj = n · xj. Hence, utility depends only on the total amount of consumption.

In the Dixit-Stiglitz framework, in contrast, there is a direct utility gain from an

increase of the number of firms/ varieties. Here, consumers profit from an increase

of the number of firms only through the reduction of mark-ups by lower market

power. We follow Blanchard and Giavazzi (2003, p. 882) in considering this effect

of market entry to be the most important.1For a more general formulation see Blanchard and Giavazzi (2003, p. 881).

3

Maximizing the utility function (1) under a budget constraint yields the de-

mand functions

Xj =βyj

Pand x0,j =

(1− β)yj

p0

(3)

where yj denotes the income of worker j, p0 is the price of the homogeneous good,

and P is the price index of the heterogeneous goods, defined by

P =

(1

n

n∑i=1

pρ

ρ−1

i

) ρ−1ρ

(4)

(see Blanchard and Giavazzi, 2003, p. 882). Income yj of a worker is either the

union wage rate wi or the competitive wage rate w0. Minimizing the expenditures

for a given value of Xj yields the following individual demand function for variety

xi:

xi,j =

(P

pi

) ρ1−ρ βyj

npi

(5)

Hence, aggregate demand for this good is

xi =

(P

pi

) ρ1−ρ β

npi

Y (6)

and depends linearly on the total income of workers Y ≡∑N

j=1 yj.

2.2 Firms

Firms in both sectors maximize profits. The homogeneous good x0 is produced

employing exclusively labor from the competitive labor market. The good serves as

a numeraire. Technology is assumed to be linear (no fixed costs), and standardized

without loss of information to x0 = L0. Market entry occurs until firms just

break even. This implies together with the assumed production function that the

competitive wage rate is unity: w0 = p0 ≡ 1. The number of firms in the perfectly

competitive sector is undetermined, but must be sufficiently large to guarantee

perfectly competitive behavior.

Each heterogeneous good is produced by a different firm, employing respec-

tively a fixed amount of ∆ units of labor from the competitive labor market. The

amount of unionized labor input can be derived from the technology constraint

Li(xi) =xi

α(7)

4

where the constant α symbolizes exogenous variable output per unionized worker.

Profit π of a representative firm reads

πi = xi · pi − Li · wi −∆

After substituting Li by the technology constraint (7) and pi by the inverse demand

function, maximization of πi yields the optimum price

pi =wi

αρor pi =

wi

α

σ

σ − 1(8)

The mark-up over marginal costs is a negative function of the number of firms n,

since ρ depends positively on n.

Market entry is free and costless. Firms enter/ exit the market until the

profits of an additional firm would be negative, and profits of all incumbent firms

are strictly nonnegative. In a symmetric equilibrium all firms i 6= 0 are equal

(xi = x, pi = p, Li = L, wi = w and πi = π = 0).

2.3 Trade unions

Assuming that workers are distributed evenly across all firms in the heterogeneous

sector2, the probability of a worker to receive the union wage rate is nL/N . The

remaining workers must work for the competitive wage rate. Each trade union

maximizes the expected utility of a representative worker, and bargains with a

fraction γ over the wage rate w (“right-to-manage model”). The number of unions

is thus 1/γ.

Given our assumptions the expected utility of a representative worker is

U+ =nL

Nuj(w,P

+) +

(1− nL

N

)uj(1, P

+)

if there is an agreement with the firms and

U− = uj(1, P−)

2If workers were distributed unevenly, some of them could increase the probability of anemployment by reallocating themselves to a firm where less workers are attached.

5

if there is no agreement.3 P+ and P− represent the price index of the heteroge-

neous goods, respectively in the cases of an agreement and of no agreement. The

bargaining parties thus take into account that the price index differs in these two

cases.

2.4 Timing of the model

Since our model is static, there is no chronological order of decisions, actions, and

reactions. But, by assuming a specific informational status of the workers, firms

and unions, we determine what may be called a logical order.

One of the main differences to Blanchard and Giavazzi (2003) is the way

market entry is modelled. In their paper, firms face entry costs that play a similar

role as the fixed costs do in ours. But since these costs are sunk costs, it is difficult

to explain why the number of firms should shrink after a marginal deterioration of

their economic situation. Blanchard and Giavazzi (2003, p. 891) argue that “firms

which die are not replaced”. But it remains open why these firms should die in

the model as long as profits are strictly positive. In our model all firms that enter

the market actually have to bear the fixed costs. Thus, starting from a zero profit

equilibrium, a deterioration of the firms’ situation leads to losses, pushing some

firms out of the market. But - as the entry costs in the Blanchard and Giavazzi

framework – fixed costs do not affect the wage bargain, if they arise independently

of whether there is an agreement or not.

From these considerations the following logical order results: i) First market

entry/ exit decisions are taken. ii) Then fixed costs arise for those firms that have

entered the market. iii) Wage bargains take place independently of each other.

It must be assumed that unions and the corresponding firms know the resulting

wages, prices and employment from all other bargaining units in the economy,

e.g. through a heuristic process, which is terminated in the long-run equilibrium

we look at. This common assumption allows us to abstract from the strategic

interplay between different bargaining units. Once wages are determined, iv) goods3Since all agents fare better in the case of an agreement, this second term serves only as the

’conflict point’ during the bargain, but is never realized.

6

are produced, sold and consumed. In a long-run equilibrium only those firms that

can actually cover fixed costs enter the market, however.

2.5 The wage bargain

Unions and firms take into account the aggregate demand functions. They are

equally aware of the responses of employment, workers’ income and prices regard-

ing changes of the wage rate. In contrast, they take the number of firms in the

heterogeneous sector as given, because it is determined “before” the bargaining.

The Nash product describing the asymmetric bargaining problem is

NP = γn(px− Lw) · (U+ − U−)δ (9)

where δ denotes the relative bargaining power of the union (Nickell, 1999, p. 3).

Fig. 1 illustrates how the negotiated wage changes due to adjustments of the number

of firms. πv ≡ γn(px−Lw) are the participating firms’ profits before the deduction

of fixed costs, and Θ ≡ (U+−U−)δ is the additional utility the union obtains in case

of an agreement, weighted by the relative bargaining power of the union. πv clearly

depends negatively on the wage rate w. In comparison, the effect of a higher wage

rate on Θ may be positive or negative. Starting from a wage rate close to zero, the

union welcomes wage increases unambiguously, since all workers remain employed

in the primary sector, and utility of each worker is boosted. But if the wage rate

surpasses a certain point, higher wages for employed workers have to be weighted

against a loss of income for those who lose their job in the primary sector. Hence,

the negotiated wage has to be between the competitive wage rate, corresponding

with point C, and the one the union would chose unilaterally (monopoly union,

corresponds with point M in fig. 1), where the positive and the negative effects of

an incremental wage rise exactly match each other. For a given curve P , a point

further to the right-hand side is associated with a lower wage rate.

The curves P 0, P 1 and P 2 depict which combinations of πv and Θ are feasible

in principle, while the hyperbolas NP 0, NP 1 and NP 2 stand for different given

values of the Nash product (9). A solution to the Nash bargaining problem corre-

sponds with a maximum of the Nash product, depicted by points Q0, Q1 and Q2.

7

-

6

w ↓ -

Θ

γnπv

P 0

P 1

P 2

•C

•0

•M

NP 0

NP 1

NP 2

• Q0

•Q1

•Q2

γn∆

Fig. 1: Nash’s bargaining solution with free market entry

Imagine that curve P 0 represents the set of combinations of πv and Θ that are

feasible initially. The negotiated wage rate corresponds to point Q0. Since firms’

revenues exceed fixed costs, newcomers are attracted, which reduces the prices of

the goods, so that at any wage rate profits are lower. Feasible combinations are

now represented by the curve P 1. The process of shrinking of the set of feasible

combinations continues until the symmetric firms just break even with the wage

rate that results from the bargaining (point Q2). Then, a long-run equilibrium is

reached.

From the union’s and the corresponding γn firms’ point of view, the hetero-

geneous goods’ price index depends on the agreed wage rate because the goods

prices depend on the wage rate and the number of covered workers is not negligible

relative to the entire economy. If we differentiate between covered and non-covered

workers and the related variables4 (in the latter case pi, Li and wi carry a bar, sym-

bolizing that these variables are regarded as being given by the bargaining parties),

(4) becomes4As noted earlier, wages, demand, labor input etc. are the same for all firms in a symmetric

equilibrium (ex post). Nevertheless, the bargaining parties consider these variables to depend onthe result of the bargain ex ante if they are related to them, and as exogenous if they are relatedto other firms/ workers.

8

P =

[1

n

(γn∑i=1

pρ

ρ−1

i +n∑

i=γn+1

pi

ρρ−1

)] ρ−1ρ

(10)

Since workers are distributed evenly across all firms, the number of workers per firm

is N/n. The probability of an employment at the union wage rate is thus Li/(N/n)

for workers who are member of the considered union, and Li/(N/n) for all other

workers. Hence, the expected income of a worker equals (nLi/N)wi +[1−(nLi/N)]

for members of this union and (nLi/N)wi + [1− (nLi/N)] for all workers that are

members of other unions (notice that the competitive wage rate is unity). Total

income from the point of view of the bargaining parties is

Y =

γN∑j=1

[nLi

Nwi +

(1− nLi

N

)]+

N∑j=γN+1

[nLi

Nwi +

(1− nLi

N

)]

=

γN∑j=1

[nLi

N(wi − 1)

]+

N∑j=γN+1

[nLi

N(wi − 1)

]+N (11)

The resulting wage rate is not amenable to a formal representation in general.

Only in the benchmark case of decentralized bargaining (γ → 1/n) a closed form

can be found, which is

wi|γ→ 1n

=δ + ρ

ρ(1 + δ)(12)

All other variables follow from the wage rate in a straightforward manner. The

effects of variations of the unions’ relative bargaining power δ and transparency ζ

are derived analytically in subsection 4.2. Except from the special case of decen-

tralized bargaining, numerical methods are appropriate to solve for the wage rate

and all other endogenous variables. This exercise is carried out in section 3 Since

ρ depends on the number of firms n, the latter has to be derived before we can

determine the wage rate. Other variables that are related to the macro level are

the price index P and total income Y .

3 The union wage gap

In this section, we solve the model developed in the previous section numerically.

Employing statistical data to distinguish a set of countries, we are able to derive a

9

union wage gap which is relatively close to estimates from the empirical literature.

If we would restrict the analysis to decentralized bargaining (γ → 1/n), it could

also be executed analytically. Yet, this parameter turns out to be a crucial one.

3.1 The data

A lack of data and the objective to work out differences between countries in a

stylized and focused fashion forced us to restrict the numerical analysis to seven

countries: The United States, the United Kingdom, West Germany, Denmark,

Canada, Italy and Japan. These countries have been chosen because the necessary

data have been available at least for some years, and because they are quite different

from each other with regard to the degree of centralization of the wage bargains

and the union coverage rate. If the data were available, we specified the parameters

for the years 1980, 1985, 1990 and 1994.

The following parameters are chosen to distinguish a country’s specific situa-

tion at different points in time: union coverage, the size of the workforce, and “the

degree of centralization of the wage bargains”, which correspond with the expen-

diture share of the heterogenous good (β), the number of workers (N), and the

fraction of covered workers who are member of one union (γ). Although the latter

is relatively stable over time (Kenworthy, 2000, p. 13), we account for variations

of it because the considered time span is fairly long, and because γ affects the

endogenous variables strongly. It is needless to say that these three parameters

cannot give a sound impression of a country’s economic situation. Yet, it turns out

that they suffice to explain much of the differences in the union wage gap between

the included countries.

We adjust β until the share of workers that are covered coincides with the bar-

gaining coverage rate, taken from Traxler (1996, p. 274), supplemented by OECD

(1997, p. 71), whenever the records were comparable.5 That is to say, we derive5Unfortunately, the definition and measurement of bargaining coverage is not unambiguous.

One difference between the reported coverage rates is that some of them adjust for the fact thatin several countries not all workers have the legal right to bargain. From the role of the parameterin the model is is clear that we must take the unadjusted coverage rate. Therefore, the OECDdata were only viable for those countries where all workers have the right to bargain, so that both

10

the decrease of union coverage in many countries from an assumed relative decrease

of the consumers’ valuation of goods produced in the unionized sector. This means

that we abstract from many causes that may have influenced the coverage rates,

too, e.g. the political environment, legislative measures etc. Examples for the shift

of preferences away from the unionized sector’s goods are numerous and include

sectors like ’steel’ and ’public transports’. There are several reasons why we draw

on union coverage rather than union membership (density). First, recent evidence

suggests that there is no union membership non-membership wage gap among the

covered employees. Measured differences between the pay of trade union mem-

bers and non-members seem to be caused by various unobservable variables, which

cause e.g. a concentration of members in high paying workplaces (see (Booth and

Bryan, 2001)). Second, the figures for coverage account for all sort of institutional

and legislative differences between the countries in the sample. For instance, in

Germany all workers whose employer is member of the employers’ association are

covered, regardless whether they are member of a union or not. In many countries,

union wages are legally extended to cover non-union firms. In these cases union

density would not reflect the number of workers who are affected by collective con-

tracts. Therefore, union coverage is the proper concept if one wishes to measure

the true impact of unionism across different countries.

The most difficult decision is regarding the appropriate measure of γ, the

degree of centralization. There is an abundance of qualitative indicators designed

to describe it (for a comprehensive survey see Kenworthy, 2000). In addition, some

authors claim that coordination rather than centralization would be the appropriate

measurement (Soskice, 1990). For several reasons, we chose to take Iversen (1998)’s

indicator of wage bargaining centralization. First, it is available annually for all

countries that we included.6 Second, the Iversen indicator takes account of small

changes towards a more centralized or decentralized wage setting. In comparison,

other indicators, like the one published by the OECD, are much more abrasive. It

must not be concealed, however, that (with the exception of Canada and the U.S.)

there are considerable differences between the alternative indicators (Kenworthy,

rates coincide. For instance, this is the case in Italy.6The 1994 value has to be taken from 1993, which is the last one published.

11

2000). Therefore, our results are not robust to the choice of indicator. We displaced

the origin of the Iversen indicator such that the smallest value, corresponding

with firm-level bargaining, is zero.7 N is civilian employment, taken from the US

Department of Labor (2002, p. 11).8 The 1980 value is standardized to unity,

respectively. It should be noted that this parametrization does not account for

variations in the number of hours worked per employee.

Why have these three parameters been chosen to characterize the countries

in the sample? First, it turns out that one parameter does not suffice to describe

the bargaining setting adequately. For instance, bargaining takes place at the firm

level in the U.S. as well as in Canada. But union coverage is significantly higher

in Canada, which yields a different outcome. The same applies if one compares

the situation of the U.S. with Japan. Union coverage is roughly comparable, but

bargaining is more centralized in Japan, leading to a noticeable lower union wage

gap. Second, this contribution sets out to derive the union wage gap from features

of labor markets and product markets. Therefore, it is desirable that at least one

parameter is included, which is also related to the product markets (i.e. N , the

’number of workers/ consumers’). This permits to examine if and to which extent

the size of the markets has an impact on the wage rate for a given structure of the

labor markets.

In contrast, we chose not to vary the production technology across countries

and time (parameters α and ∆ in the model). Even though these parameters play

an important role for the level and dynamics of real wages and income, we abstract

from variations in them because our focus is on the comparative-static effects of the

bargaining structure and the size of the unionized sector. In addition, it is difficult

to obtain reliable data on costs. The latter is equally valid for the parameters

δ (relative union bargaining power) and ζ (determining how strong the number

of varieties affects the elasticity of substitution). We chose to employ the same

parameter values for ζ and δ for each point in time and country because a lack of

data would otherwise make the outcome additionally arbitrary.7To avoid computational problems, a value of 0.0001 rather than literally zero is the minimum

(employed for the U.S. and for Canada).8This source converts the national data such that they approximate U.S. concepts. The Danish

values stem from .

12

Table 1 specifies the parameter values employed, where union coverage, sym-

bolized by ψ, is given in addition to the corresponding values of β. α is standardized

to unity for the ease of computation. Fixed labor input ∆ is 0.002, which causes a

ratio of fixed costs to total costs (cost disadvantage ratio, CDR) within the range

13.6%–24.5%.9 Furthermore, we specify the parameters δ and ζ as 1 and 0.1, re-

spectively, implying symmetric bargaining and a relatively weak responsiveness of

the elasticity of substitution with regard to market entry.

Table 1. Parameter specification

Canada Denmark Germany Italy

year 1985 1990 1994 1980 1990 1994 1980 1985 1990 1980 1990 1994

β 0.491 0.494 0.472 0.823 0.823 0.824 0.901 0.882 0.897 0.996 0.972 0.965

ψ 0.37 0.38 0.36 0.69 0.69 0.69 0.76 0.74 0.76 0.85 0.83 0.82

γ 0.000 0.000 0.000 0.404 0.257 0.329 0.242 0.243 0.249 0.071 0.071 0.194

N 1.061 1.194 1.200 1.000 1.068 1.015 1.000 0.982 1.055 1.000 1.044 0.987

Japan U.K. U.S.A. Each country/year

year 1980 1990 1980 1990 1994 1980 1985 1990 1994 α 1.000

β 0.352 0.293 0.842 0.595 0.598 0.372 0.298 0.270 0.268 ∆ 0.002

ψ 0.25 0.21 0.70 0.47 0.47 0.26 0.20 0.18 0.18 δ 1.000

γ 0.160 0.265 0.052 0.052 0.052 0.000 0.000 0.000 0.000 ζ 0.100

N 1.000 1.130 1.000 1.084 1.043 1.000 1.079 1.196 1.239

3.2 Results

Figure 2 shows the relative differences between the union wages and the competitive

wage rates in the model. The highest wage gap is found in the U.S., where it equals

14.7% in 1980, increases to 16.3% in 1985 and then decreases to 15.9% in 1994.

Canada features the second highest value, which is roughly 11.5%. The U.K. wage

gap increases from 8.2% in 1980 to 9.7% in 1994. In contrast, the union wage gap

decreases in Japan from about 10.5% in 1980 to 7.2% in 1990. In the time span,9These values are comparable with those of Elbehri and Hertel (1999).

13

the Danish union wage gap increases from 3.0% to 5.2%, and then decreases to

4.2% in 1994. The German and Italian values remain nearly constant from 1980

to 1990 (5.4% and 7.1%, respectively). In 1994 the Italian wage gap falls to 5.9%,

whereas the German figure could not be calculated because of missing data.

union wagegap [%]

0

5

10

15

1980 1985 1990 1995

Canada Denmark Germany Italy

Japan U.K. U.S.A.

Fig. 2: The union wage gap

Decentralized wage bargaining apparently explains the high wage gap in Canada

and the U.S. Unions disregard the negative effect higher wages have on the aggre-

gate price level, since the number of represented workers is small relative to the

total workforce. The inverse accounts for Denmark, where wage bargains concern

a large fraction of the workforce, and the wage gap is the lowest. Danish unions

internalize the negative effect higher wages have to a great extent. This effect is

well explored in the literature (for a short summary of various external effects that

play a role see Boeri, Brugiavini and Calmfors, 2001), and it causes much, yet not

all of the differences between the countries’ development of the wage gap.

14

Since Canada and the U.S. have both decentralized bargaining, and relative

variations of the total workforce are similar, differences in union coverage explain

why Canada’s wage gap is roughly 4 percentage points lower. But why should

higher union coverage yield a lower wage? Usually, it is taken for given that cover-

age is a proxy for a union’s bargaining power, which is supposed to have a positive

effect on the negotiated wage rate. In our framework, in contrast, a higher coverage

rate is caused by a higher expenditure share of the unionized sector of the economy,

so that there are more monopolistically competitive firms. This implies that the

heterogeneous goods become closer substitutes (ρ increases) so that the optimum

prices and the firms’ ability to accrue rents diminish. Therefore, union wages de-

crease. The moderate increase of the wage gap in the U.K. can also be explained

by changes of the expenditure share. In all other countries the expenditure shares

remained relatively stable.

The decrease of the unionized sector in the U.S. in the considered time span did

not lead to a relevant modification of the wage gap, however, because the relatively

strong increase of the total workforce works against this effect. More workers lead

to a higher number of firms, which enhances competition in the goods markets.

This reduces rents and union wages. In Canada, this effect yields the moderate

decrease of the wage gap from 1985 to 1990. For all other countries, the variations

of the total workforce are small relative to the variations of union coverage and the

degree of centralization.

What do our results imply for real income? If we deflate wages by the consumer

price index (15), we find that the highest real wages in the primary sector occur in

the U.S. Here, and in the U.K., we also observe the strongest increase of the real

wage rate during the 1980s. The reason is, besides the modest increases of union

wages in these countries, that the expenditure share β of the goods produced by

firms that face union wage bargaining has dramatically declined in the considered

time span. This lowers the cost-of-living price index (15) and raises real wages. In

the U.S and in the U.K. ever less workers profited ever more from the existence of

trade unions. Why this happens seems a question worth being investigated.

15

3.3 Comparison with empirical evidence

There is no lack of empirical work on the union wage gap. The problem is rather

that the existing empirical literature is partially inconsistent because of differences

with regard to the employed data and methodology. Therefore, it is advisable

to keep at one source to increase comparability. We chose Blanchflower (1996),

supplemented by Blanchflower and Bryson (2003), since it is the only study we

found that includes nearly all countries for which we parameterized the model in

the considered time span (except from Denmark). The underlying data from the

International Social Survey Programm Series (ISSP, 1985–1993) does not allow to

control for important variables such as industry, which is likely to bias the estima-

tions upwards. More reliable are the results in Blanchflower and Bryson (2003),

using the MORG files of the CPS for the U.S. and the British Social Attitudes

Surveys (BSAS) for the U.K., which we averaged for the time span 1985–1993

to make them comparable. Table 2 contrasts the estimates of Blanchflower and

Bryson (2003) for the U.K and the U.S., and Blanchflower (1996) for the rest of

the countries with our model’s results. For Italy, the U.K. and the U.S. the differ-

ences appear small. The estimated wage gap for Japan (47.8%) is much higher than

what the model predicts (and also much higher than what we think is plausible).

The estimates for Germany (3.4%) and Canada (4.8%) are lower than what our

model predicts. However, Blanchflower and Freeman (1992) find a wage gap of 6%

for West Germany, which again is nearby the model’s result. Blanchflower (1996,

p. 23, footnote 17) cites several studies that estimate wage gaps for Canada which

are similar to the outcome of our model. Altogether the model’s results seem to

be relatively close to the estimated wage gaps, in particular in those cases where

the latter are reliable.

One reason why estimated wage gaps may not be reliable is that empirical

studies frequently use union density (membership) as a measure of union influence.

This may be reasonable for the U.S., Japan and Canada, where union density and

coverage are roughly equivalent. In Continental Europe substantial gaps between

union density and coverage prevail, however. An extreme case is France, where

union membership is about 10%, and where coverage of collective agreements is

16

Table 2. Union wage gap in the model and empirical evidence

Model Estimates

1980 1985 1990 1994 85–93

Canada 11.9 11.0 11.3 4.8a

Denmark 3.0 4.1 5.2 4.2

Germany 5.4 5.5 5.2 3.4

Italy 7.2 7.2 7.1 5.9 7.2

Japan 10.5 8.8 7.2 47.8

U.K. 8.2 8.9 9.6 9.7 7.3

U.S. 14.7 16.3 16.2 16.0 19.2a not significantly different from zero

Sources: Blanchflower (1996), Blanchflower and Bryson (2003), own calculations

about 95% (see Visser (2003)). This specification biases the estimated union non-

union wage gap downwards, which may partially explain the difference between the

simulated and the estimated wage gap for Germany. Another important reason for

differences between our results and some estimations is selection. Longitudinal

evidence has shown that there is positive selection into unions among low-skilled

workers, and negative selection among high-skilled workers (Hirsch, 2004), which

biases the measured effect of unions on wages. In some countries, however, the

results of the bargains are de facto or even legally extended to workers who are not

member of a union. In these cases selection can only have a very limited impact.

How robust are our results? First, we checked the sensitivity of the results

with respect to variations of the exogenous parameters α, ∆, δ and ζ. None of

them has a strong impact on the union wage gap. Therefore, we conclude that

the parameters that describe the countries in the numerical analysis (β, γ and N)

are indeed decisive for the union wage gap. In contrast, employment, the elasticity

of substitution, prices, and the number of firms depend more on the choice of

the former parameters, which determines e.g. real income. Therefore, an analysis

aiming at explaining the course of real income would have to account for their role

more accurately than we do.

17

A second possibility to check our results is to compare computed variables

(other than the wage gap) with estimates in the literature. We perform such a

comparison with respect to the markup of prices on marginal costs, taken from

Oliveira Martins, Scarpetta and Pilat (1996). Before we turn to the results of this

comparison, some words of caution are in order, however. First, the employed

method of estimation, put forward by Roeger (1995), produces a downward bias

with increasing returns to scale. Thus, the estimates are likely to represent a

lower bound. Second, the estimated markups are the average of sectoral markups

in the period 1980-1992, weighted by 1990 production shares in manufacturing.

Therefore, they can only be used to give a rough impression of the differences

between countries.

Table 3. Markups on prices in the model and empirical evidence

Model Estimates

Canada 23 20

Denmark 18 15

Germany 17 21

Italy 16 19

Japan 32 26

U.K. 20 15

U.S.A. 32 14

Source: Oliveira Martins et al. (1996), own calculations

The third column of table 3 stems from Oliveira Martins et al. (1996, p. 25).

The second column averages the markups we obtain in the numerical analysis

for the respective countries at different points in time. Taking into account that

the estimated markups are probably biased downward because of the implicitly

assumed constant returns to scale, the results for Canada, Denmark, Japan and

the U.K. are supported by the proving. The computed markups for Germany and

Italy are quite close to the estimated values, but smaller than the latter. The

ranking of the U.S. values is as supposed, but they display the largest differences.

One possible explanation why the computed markup for the U.S. is too high is that

here the neglected perfectly competitive sector with zero markup is the largest.

18

In summary, the following results seem to be fairly stable:

• The union wage gap largely depends on the degree of centralization of the

bargains, and, to a somewhat lesser extent, on the expenditure share of the

unionized sector’s goods and on the size of the employed labor force. At odds

with a widely held view, the latter two have a negative effect on the wage

gap because competition on the goods market is reinforced, which reduces

the bargained wage rate.

• In contrast, the bargaining power of unions, commonly regarded as important

explanatory factor, turns out to have only a limited influence on the union

wage premium.

• Differences between countries with respect to real income per worker can

partially be explained by the expenditure share of the unionized sector.

The fact that our model can explain some differences between the wage gaps

of various countries and at various points in time leads us to the conclusion that it

indeed covers some of the most important features of imperfect product and labor

markets. Therefore, it is quite natural to ask how policy may influence the outcome

according to the respective goal pursued, which is done in the following section.

4 Regulation of labor and goods markets

The basic theoretical model described in section 2 comprises of the equations that

determine the wage rate w, variable employment per firm L, the product price

p, demand for each variety x, number of firms n, the total income Y , the price

index P and the elasticity of substitution σ. These endogenous variables depend

on the number of workers N , the expenditure share of the heterogeneous goods β,

output per unionized worker α, the fixed labor input ∆, and on the parameters: δ,

γ, and ζ. The latter can be thought of as being determined at least partially by

policymakers:

19

ζ The parameter ζ impacts on the responsiveness of the elasticity of substitu-

tion between any two varieties, σ, with respect to market entry. This param-

eter is referred to as ‘transparency’, because only if a market is sufficiently

transparent, the number of competitors impacts on consumer behavior. Ad-

mittedly, this definition is much narrower than the vernacular meaning of

‘transparency’. But in the context of our model, we do think that trans-

parency may actually be seen as the ability of households to perceive and

compare the supply of goods by different firms. How can economic policy

influence this ’transparency’? One instance are laws that bind firms on pub-

lished prices or force firms to comply with specific norms (e.g. DIN or ISO

standards). Although this is only a terminological difference, it should be

noted that this interpretation is rather the inverse of the one Blanchard and

Giavazzi have in mind for their related parameter σ, where increases of the

parameter are named “product market deregulation”. In our interpretation

transparency and substitutability are enhanced rather than worsened by reg-

ulatory measures.

δ This parameter symbolizes the relative bargaining power of trade unions. If

its value is zero, wages are competitive. The case δ → ∞ means that the

union can set wages unilaterally (monopoly union). Pencavel (2002, p. 15ff.)

depicts among other things how the Thatcher administration depressed union

power in Britain. Examples of policy measures include the legislation on

strikes, lock-outs, union governance and the closed shop.

Although γ is a very important parameter in the model (see section 3), which

may also be influenced by policy measures, we have to exclude it from the analysis.

The reason is that a positive value of γ renders the model so complicated that it

cannot be solved analytically.

We now derive how variations of ζ (product market transparency) and δ (union

bargaining power) affect the equilibrium values of some selected variables. We as-

sume decentralized bargaining (γ → 1/n), which is the standard in the bargaining

literature. We calculate the responses of the equilibrium values of selected en-

dogenous variables to variations of the policy parameters ζ and δ. In doing so,

20

we focus on the union wage rate, and on the variables that are determined at the

macro-level, i.e. aggregate income Y , the price index P , the number of firms n,

and, derived from the latter, the degree of substitutability, represented by σ.

4.1 Equilibrium

In a symmetric equilibrium with decentralized bargaining, many analytical expres-

sions simplify considerably. In particular, the wage rate is given by (12), which

still contains the endogenous σ, however. In order to keep the results simple and

manageable, we do not substitute all endogenous variables. Instead, we try to dis-

play all results as compact as possible. In a symmetric equilibrium the price index

(4) becomes

P = p1 = p2 = . . . = p

If P = pi, demand for one heterogeneous commodity (6) becomes

xi =βY

nP(13)

Hence, from the technology constraint (7)

L =xi

α=

βY

αnP(14)

The cost-of-living price index P̂ can be derived by a weighting of the prices in

both sectors with the respective expenditure shares:

P̂ = P β · 11−β = pβ (15)

In the symmetric case, aggregate income (11) becomes

Y = nL(w − 1) +N (16)

As in the Dixit-Stiglitz framework the number of firms/ heterogeneous goods

is determined through the assumption that firms’ profits are zero in equilibrium.

Setting πi = 0, making use of equations (14) and (8), we get (1 − ρ)px − ∆ = 0.

With equation (13), and from the definition of ρ, we know that 1 − ρ = 1/(ζn).

Employing this information, the zero-profits condition yields:

n =

√βY

∆ζ(17)

21

The three equations for the price index P , total income Y and the number of

firms n determine - together with the results and definitions derived before - the

simultaneous long-run equilibrium. For convenience, we now recapitulate the most

important results derived in section 2:

σ = ζn > 1 (2) (18)

pi =wi

α

σ

σ − 1(8) (19)

wi|γ→ 1n

=δ + ρ

ρ(1 + δ)(12)

= 1 +δ

(σ − 1)(1 + δ)(20)

It is possible to verify the correctness of the model’s outcome by the redundant

equilibrium condition of the homogeneous market: xdemand0 = (1 − β)Y = N −

n(L + ∆) = xsupply0 . The following variables are derived endogenously: wage rate

w, variable employment per firm L, product price p, demand x, number of firms n,

total income Y , price index P , and elasticity of substitution σ. Other variables, like

the demand for the homogeneous good may be deduced from them. The results

depend on the total number of workers N , output per unionized worker α, the

expenditure share of the heterogeneous goods β, the unions’ relative bargaining

power δ, the fixed labor input ∆, the degree of centralization of the bargain γ, and

on the parameter ζ, which indicates how strong competition on the goods market

is affected by market entry.

Making use of (19) and inserting the wage rate (20), we obtain for the het-

erogenous price index

P =(δσ + σ − 1)σ

α(σ − 1)2(1 + δ)(21)

Inserting the wage rate (20) and the number of employed workers per firm (14)

into nominal income (16), we get

Y =βY

αP

δ

(σ − 1)(1 + δ)+N

Solving this equation for Y , and substituting P by means of equation (21) obtains

Y =(δσ + σ − 1)

δσ + (σ − βδ)(σ − 1)/σN (22)

22

In the following subsection, we assess how variations of the policy parameters

δ and ζ affect the equilibrium described by the above set of equations.

4.2 The effects of goods and labor market regulation

The policy parameters have direct and indirect effects on the endogenous variables.

In abbreviated form, the functional relationships are:

P = P (δ, σ); Y = Y (δ, σ); n = n(Y, ζ); σ = σ(ζ, n); and w = w(δ, σ) (23)

For instance, to calculate the total effect of an increase of the union bargaining

power δ on aggregate income Y , we have to take into account the direct effect of δ

on Y , which is given by the partial derivative, and the effect of δ on substitutability,

σ, which also impacts on Y indirectly. Partial and total derivatives of variable vi

with respect to variable vj are denoted ∂vi/∂vj and dvi/dvj, respectively. The total

differential of Y from equation (23) reads

dY =∂Y

∂δdδ +

∂Y

∂σdσ

where, if we hold ζ constant,

dσ =∂σ

∂ndn

and

dn =∂n

∂YdY

which yields the total derivative

dY

dδ=

∂Y∂δ

1− ∂Y∂σ

∂σ∂n

∂n∂Y

(24)

In order to derive the total effects of variations in ζ and δ, we first evaluate the

partial derivatives of w, P , Y , n and σ with respect to the endogenous variables

and the policy parameters. These partial derivatives are given in the appendix.

Furthermore, we define:

Definition 1

Substitutability between goods is high if (σ − 1)2 + δσ(σ − 2) > 0, and is low

23

otherwise. This implies that, if σ ≥ 2, substitutability is always high, and if σ → 1

it is low.

It should be noted that high substitutability is the more likely case, since, for

substitutability to be low, σ must be considerably below 2. Employing definition 1

and the partial derivatives given in the appendix, we derive the following results

for the case of decentralized bargaining:

Proposition 1

Aggregate nominal income, the number of firms, and substitutability between goods

depend positively on union bargaining power δ.

Proof: Using equation (24) and the partial derivatives given in the appendix,

the total derivative of Y with respect to δ can be expressed as

dY

dδ=

2Nβσ(σ − 1)2(δσ + σ − 1)

δ [σ2 − β(σ − 1)] + σ(σ − 1)(25)

· 1

2 δ2σ3 + δσ2 [4 (σ − 1)− βδ] + (σ − 1)2(2σ − βδ)> 0

The expression has been rearranged such that the content within every bracket has

a strictly positive sign. One can then see that the total effect is unambiguously

positive.

The direction of the effect of a change in the union bargaining power on the

number of firms is unambiguous and yields from the signs of the total derivative of

Y and the partial derivative given in the appendix. It can be expressed as

dn

dδ=∂n

∂Y

dY

dδ> 0 (26)

What is the economic reason for this result? Intuition would suggest that the

strength of a union impacts negatively on the profits of firms so that the result-

ing equilibrium comes along with fewer instead of more firms. The result can be

explained by a sort of collusion between the bargaining parties. Wage increases

lead to proportional increases of the firms’ prices. Since the competitive wage rate

remains unchanged, the firm’s total costs rise less than proportional. Thus, covered

workers and firms profit to the disadvantage of workers who are not covered by the

collective agreement.

24

Substitutability is measured by σ or ρ, which depend directly on transparency,

ζ, and on the number of firms, n. Therefore, employing eqs. (26) and the partial

derivative given in the appendix, and holding ζ constant, the effect of an increase of

the relative union bargaining power, δ on σ, obtains as the unambiguously positive

total derivativedσ

dδ=∂σ

∂n

dn

dδ> 0 (27)

�

Proposition 2

Product market transparency ζ impacts positively on substitutability between

goods. The effect on nominal income depends on whether substitutability is high

or low (see definition 1). By contrast, increasing transparency unambiguously de-

creases the equilibrium number of firms.

Proof: The total derivative of σ with respect to transparency, ζ, and holding δ

constant, isdσ

dζ=

∂σ∂ζ

+ ∂σ∂n

∂n∂ζ

1− ∂σ∂n

∂n∂Y

∂Y∂σ

Substitution of the partial derivatives given in the appendix and rearranging yields

dσ

dζ=

{[σ2 − β(σ − 1)] δ + σ(σ − 1)} (δσ + σ − 1)n

δ2σ2(2σ − β) + 4 δσ3(σ − 1) + (2σ − βδ)(σ − 1)2 > 0

Since the content of each bracket is positive, it the whole expression is positive,

too. The result does not hinge on the degree of substitutability.

The total derivative of Y with respect to ζ is

dY

dζ=∂Y

∂σ

dσ

dζ(28)

The sign of the partial derivative ∂Y/∂σ is ambiguous in general. Economically, this

is caused by the following trade-off: On the one hand an increase in σ decreases the

mark-up, which increases demand and, thereby, augments employment and total

income. On the other hand, wages decrease through the reduction of rents, which

impacts negatively on total income. If σ is high, the latter effect prevails, ∂Y/∂σ

is negative (see eq. A1 in the appendix). Hence the total derivative has a negative

sign. Since the scope for ’low substitutability’ is defined quite narrowly, this is the

most likely case.

25

The derivative of the number of firms with respect to market transparency isdn

dζ=∂n

∂ζ+∂n

∂Y

dY

dζ(29)

After substituting the partial derivatives given in the appendix, with σ = 1/(1−ρ),

and the total derivative of Y , (28), the expression becomesdn

dζ= −{[1− βρ(1− ρ)]δ2 + δρ(2− β) + ρ2 + βδρ2(2− ρ) + βρ3(1− ρ)}

2δ2(1− βρ) + βδρ2(1 + 2δ) + βρ3(1− ρ) + δρ(4− β) + 2ρ2

·n2(1− ρ) < 0

Again, the terms have been grouped such that the expression within every bracket

is strictly positive. Accordingly, the entire derivative is negative. This result is

independent of whether substitutability is high or not, hence the first term on the

right-hand side of eq. (29) (the partial derivative of n), which is clearly negative,

eliminates the ambiguity of the second term, which stems from the possibly positive

effect on nominal income. �

The result concerning variations of market transparency is to be expected,

whereas it may be surprising that stronger unions increase substitutability in the

model. The reason is that stronger unions increase total nominal income, which

has a positive effect on the number of firms.

Proposition 3

The union wage and the price-index of the heterogeneous goods depend positively

on the union bargaining power and negatively on product market transparency.

Proof: See the appendix.

Clearly, the effects on pi and on the cost-of-living price index P̂ have the same

direction as those on P . Deriving the effects on real income, i.e. on total nominal

income, deflated by the cost-of-living price index, is somewhat more demanding,

however. Since both, cost-of-living price index and total nominal income depend

positively on the bargaining power of the unions, and negatively on product market

transparency, it is interesting to see which effect on real income prevails.

Proposition 4

Total real income depends negatively on the bargaining power of the unions and

depends positively on product market transparency.

Proof: See the appendix.26

The latter results are unambiguous. In particular, they do not hinge on the

degree of substitutability. It should be noted that not all workers fare worse with

stronger unions. Real income of a primary-sector worker increases due to an in-

crease of the bargaining power of the corresponding union. But real income of a

secondary-sector employed falls, since his nominal income remains the same, while

the price-index rises. The inverse reasoning applies for the effects of an increase of

transparency.

Proposition 4 is particularly interesting, because real income is the natural

target of interventions in the model. It suggests that a government that grasps at

increasing real income should impair unions and foster market transparency. In

the model, such a combined policy would have a number of side-effects, however.

Most importantly, both measures impact negatively on the number of firms in the

heterogeneous sector of the economy. Nominal income, wages, and prices would

decrease. The effect on substitutability is undetermined, since the decrease of union

power impacts negatively, and the increase of transparency impacts positively on it.

Proposition 4 implies that unions disregard the negative impact of wage increases

on the price-level, which harms workers who are not covered by the bargain. In

other words, they partially externalize the negative effects of higher wages. In the

model, the extent to which unions take this effect into account can be varied by

the policy parameter γ, the degree of centralization of the bargain. This parameter

has been set to zero in this section, because otherwise the total effects of variations

in δ and ζ could not have been derived analytically.

5 Summary

The theoretical model developed in this paper is designed to analyze the interplay of

imperfections on product and labor markets. One sector of the products markets is

characterized by monopolistic competition, while the other is perfectly competitive.

The labor market is dichotomized as well: some workers receive the higher wage

rate determined by collective agreements, and some receive the competitive wage

rate.

27

Our approach proves fruitful in explaining the so-called ’union wage gap’, i.e.

the difference in pay between workers who are or are not covered by collective

bargaining. Employing statistical data characterizing some chosen countries, we

numerically calculate the union wage gaps and compare the results with recent

empirical estimates. In general, these figures are quite close. This is remarkable,

since we neglected factors such as differences in the technology or the tradition

of worker-employer relationships in the model, which thus seem to be of minor

importance for the wage gap. We conclude that the model comprehends the most

important channels through which products and labor markets interact.

Since we think that the model is capable to deal with imperfections on products

and labor markets and their interactions, it is straightforward to ask how economic

policy may influence the outcome in order to improve the conditions for workers and

/ or firms. We chose two parameters which are partially determined by regulatory

/ deregulatory measures: ‘Product market transparency’ and ‘union bargaining

power’.

The former has a strong negative effect on the formation of rents by enhancing

substitutability. This causes nominal wages, prices and the number of firms to

decline, whereas real income increases. Union bargaining power impacts positively

on nominal wages and income, but negatively on total real income. It has a positive

effect on the number of firms and the degree of substitutability.

References

Antweiler, Werner and Daniel Trefler (2002). “Increasing Returns and All That: AView from Trade.” American Economic Review, vol. 92 pp. 93–119.

Blanchard, Olivier Jean and Francesco Giavazzi (2003). “Macroeconomic Effects ofRegulation and Deregulation in Goods and Labor Markets.” Quarterly Journalof Economics, vol. 118 pp. 879–908.

Blanchflower, David G. (1996). “The Role and Influence of Trade Unions in theOECD.” Discussion Paper 310, Centre for Economic Performance.

28

Blanchflower, David G. and Alex Bryson (2003). “Changes over Time in UnionRelative Wage Effects in the UK and the US revisited.” In: John T. Addisonand Claus Schnabel, (Eds.) “International Handbook of Trade Unions, Chapter7,” Edward Elgar, Cheltenham, U.K.

Blanchflower, David G. and Richard B. Freeman (1992). “Unionism in the USand other Advanced OECD Countries.” In: M. Bognanno and M. Kleiner,(Eds.) “Labor Market Institutions and the Future Role of Unions,” pp. 56–79.Blackwell, Oxford.

Boeri, Tito, Agar Brugiavini, and Lars Calmfors, (Eds.) (2001). The Role of Unionsin the Twenty-first Century. Oxford University Press, Oxford.

Booth, Alison L. and Mark L Bryan (2001). “The Union Membership Wage-Premium Puzzle: Is there a Free Rider Problem?” Discussion Paper 2879,CEPR.

Dixit, Avinash K. and Joseph E. Stiglitz (1977). “Monopolistic Competition andOptimum Product Diversity.” American Economic Review, vol. 67 pp. 297–308.

Ebell, Monique and Christian Haefke (2003). “Product Market Deregulation andLabor Market Outcomes.” Discussion Paper 957, IZA.

Elbehri, Aziz and Thomas W. Hertel (1999). “Economic Integration, Market Struc-ture and Growth Dynamics.” Paper presented at the Second Annual Confer-ence on Global Economic Analysis, Denmark.

Hirsch, Barry T. (2004). “Reconsidering Union Wage Effects: Surveying New Evi-dence on an Old Topic.” Journal of Labor Research. Forthcoming.

Iversen, Torben (1998). “Wage Bargaining, Central Bank Independence, and theReal Effects of Money.” International Organization, vol. 52 pp. 469–504.

Kenworthy, Lane (2000). “Quantitative Indicators of Corporatism: A Surveyand Assessment.” Discussion Paper 4, Max-Planck-Institut für Gesellschafts-forschung.

Nickell, Stephen J. (1999). “Product Markets and Labour Markets.” Labour Eco-nomics, vol. 6 pp. 1–20.

29

Nicoletti, Giuseppe, Andrea Bassanini, Ekkehard Ernst, Sebastien Jean, PauloSantiago, and Paul Swaim (2001). “Product and Labour Markets Interactionsin OECD Countries.” Working Paper 312, OECD Economics Department.

OECD (1997). “OECD Employment Outlook.” Paris.

Oliveira Martins, Joaquim, Stefano Scarpetta, and Dirk Pilat (1996). “Mark-upRations in Manufacturing Industries: Estimates for 14 OECD Countries.”Working Paper 162, OECD Economics Department.

Pencavel, John (2002). “The Surprising Retreat of Union Britain.” In: RichardBlundell, David Card, and Richard B. Freeman, (Eds.) “Seeking a PremierLeague Economy,” National Bureau of Economic Research, Cambridge, MA.

Roeger, Werner (1995). “Can Imperfect Competition explain the Difference betweenPrimal and Dual Productivity Measures? Estimates for US manufacturing.”Journal of Political Economy, vol. 103(2) pp. 316–330.

Saint-Paul, Gilles (1996). Dual Labor Markets: A Macroeconomic Perspective. MITPress, Cambridge, Mass.

Soskice, David (1990). “Wage Determination: The Changing Role of Institutions inAdvanced Industrialized Countries.” Oxford Review of Economic Policy, vol.Vol. 6, No. 4 pp. 36–61.

Traxler, Franz (1996). “Collective Bargaining and Industrial Change: A Case ofDisorganization? A Comparative Analysis of Eighteen OECD Countries.” Eu-ropean Sociological Review, vol. Vol. 12, No. 3 pp. 271–287.

US Department of Labor (2002). “Comparative Civilian Labor Force Statistics:Ten Countries, 1959 – 2001.” (www.bls.gov/fls/flsforc.pdf).

Visser, Jelle (2003). “Unions and Unionism Around the World.” In: John T.Addison and Claus Schnabel, (Eds.) “The International Handbook of TradeUnions,” Edward Elgar, Cheltenham.

30

Appendix

A. Partial derivatives

The wage rate is given by (20) and reads

w = 1 +δ

(σ − 1)(1 + δ)

Its partial derivatives with respect to δ and σ are:

∂w

∂δ=

1

(σ − 1)(1 + δ)2 > 0 and∂w

∂σ=

−δ(σ − 1)2(1 + δ)

< 0

The price index for heterogeneous goods equals (21):

P =(δσ + σ − 1)σ

α(σ − 1)2(1 + δ)

Its partial derivatives with respect to σ and δ are

∂P

∂δ=

σ

α(σ − 1)2(1 + δ)2 > 0 and∂P

∂σ=−(σ + 2 δσ − 1)

α(σ − 1)3(1 + δ)< 0

Aggregate nominal income can be expressed as (22)

Y =(δσ + σ − 1)σN

δσ2 + (σ − βδ)(σ − 1)

and its partial derivatives read

∂Y

∂δ=

βσ(σ − 1)2N

[δσ2 + (σ − βδ)(σ − 1)]2> 0

(A1)∂Y

∂σ=

−βδN[(σ − 1)2 + δσ(σ − 2)

][δσ2 + (σ − βδ)(σ − 1)]2

The sign of the latter derivative depends on the sign of (σ − 1)2 + δσ(σ− 2), whichis ambiguous in general. If substitutability is high (see definition 1), the term ispositive, and the partial derivative has a negative sign.

The number of heterogeneous firms is (see equation (17))

n =

√βY

∆ζ

31

which yields the partial derivatives

∂n

∂Y=

n

2Y> 0 and

∂n

∂ζ=−n2ζ

< 0

Substitutability is measured by σ:

σ = ζn

(see equation (18)). Its partial derivatives are respectively

∂σ

∂n= ζ > 0 and

∂σ

∂ζ= n > 0

B. Proof of proposition 3

The signs of the total derivatives of w and P with respect to product markettransparency, ζ, follow directly from the results of proposition 2 and the partialderivatives with respect to σ:

dw

dζ=∂w

∂σ

dσ

dζ< 0

anddP

dζ=∂P

∂σ

dσ

dζ< 0

Deriving the direction of the impact of more powerful unions on the nomi-nal union wage rate and the price-index requires a closer look because there arerespectively two opposite effects:

dw

dδ=∂w

∂δ+∂w

∂σ

dσ

dδ

anddP

dδ=∂P

∂δ+∂P

∂σ

dσ

dδSubstitution of the partial derivatives and transformations (σ = 1/(1− ρ)) yield

dw

dδ=

βρ3(1− ρ) + 2ρ2 + 2βδρ2 + 2δρ(2− β) + δ2[2− 3βρ(1− ρ)]

2δ2(1− βρ) + δρ(4− β) + 2ρ2 + βδρ2 + 2βδ2ρ2 + βρ3(1− ρ)

1− ρ

ρ(1 + δ)2> 0

anddP

dδ=

2δ2[1− 2βρ(1− ρ)] + δρ(4− 3β) + ρ2(2− β) + 2βδρ2 + βρ3(2− ρ) + βδρ3

2δ2(1− βρ) + δρ(4− β) + 2ρ2 + βδρ2 + 2βδ2ρ2 + βρ3(1− ρ)

· 1− ρ

αρ2(1 + δ)2> 0 (A2)

The terms have been arranged such that the content of each bracket is strictlypositive (note that ρ(1− ρ) ≤ 1/4, since 0 < ρ < 1). �

32

C. Proof of proposition 4

Total real income is defined as total nominal income, Y , deflated by the cost-of-living price index P̂ . The total derivative of real income with respect to unionbargaining power, δ is

dY

P̂

dδ=

dYdδP̂ − Y dP̂

dδ

P̂ 2=

1

P̂·(dY

dδ− Y β

P

dP

dδ

)Inserting the previous results in (25) and (A2), and simplifying gives

dY

P̂

dδ= −2δ2(1− βρ) + 2δ2ρ(1− β) + δρ(4− 3β) + ρ2(2− β)(1 + δ) + βρ3

2δ2[1− βρ(1− ρ)] + δρ(4− β) + βδρ2 + 2ρ2 + βρ3(1− ρ)

· βY (1− ρ)2

(1 + δ)(ρ+ δ)· 1

P̂< 0

An analogous proceeding yields

dY

P̂

dζ=

dYdζP̂ − Y dP̂

dζ

P̂ 2=

1

P̂·(dY

dζ− Y β

P

dP

dζ

)=

2δ2[(1− βρ(1− ρ)] + δρ[ρ(1− β) + (3− 2ρ2) + βρ2] + ρ2(1− ρ2)

2δ2[1− βρ(1− ρ)] + δρ(4− β) + βδρ2 + 2ρ2 + βρ3(1− ρ)

·βY (1− ρ)

ρζ· 1

P̂> 0

Since the expressions have been arranged such that the content of each singlebracket is positive, the signs of the total derivatives can easily be verified. �

33